Optimal. Leaf size=110 \[ \frac{3 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^3}-\frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{7/2}}-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac{\left (b x+c x^2\right )^{5/2}}{5 c} \]
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Rubi [A] time = 0.0350046, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {640, 612, 620, 206} \[ \frac{3 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^3}-\frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{7/2}}-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac{\left (b x+c x^2\right )^{5/2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int x \left (b x+c x^2\right )^{3/2} \, dx &=\frac{\left (b x+c x^2\right )^{5/2}}{5 c}-\frac{b \int \left (b x+c x^2\right )^{3/2} \, dx}{2 c}\\ &=-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac{\left (b x+c x^2\right )^{5/2}}{5 c}+\frac{\left (3 b^3\right ) \int \sqrt{b x+c x^2} \, dx}{32 c^2}\\ &=\frac{3 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^3}-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac{\left (b x+c x^2\right )^{5/2}}{5 c}-\frac{\left (3 b^5\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{256 c^3}\\ &=\frac{3 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^3}-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac{\left (b x+c x^2\right )^{5/2}}{5 c}-\frac{\left (3 b^5\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{128 c^3}\\ &=\frac{3 b^3 (b+2 c x) \sqrt{b x+c x^2}}{128 c^3}-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2}}{16 c^2}+\frac{\left (b x+c x^2\right )^{5/2}}{5 c}-\frac{3 b^5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.152296, size = 109, normalized size = 0.99 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (8 b^2 c^2 x^2-10 b^3 c x+15 b^4+176 b c^3 x^3+128 c^4 x^4\right )-\frac{15 b^{9/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{640 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 126, normalized size = 1.2 \begin{align*}{\frac{1}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{bx}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}}{16\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{b}^{3}x}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{4}}{128\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{b}^{5}}{256}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95037, size = 451, normalized size = 4.1 \begin{align*} \left [\frac{15 \, b^{5} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (128 \, c^{5} x^{4} + 176 \, b c^{4} x^{3} + 8 \, b^{2} c^{3} x^{2} - 10 \, b^{3} c^{2} x + 15 \, b^{4} c\right )} \sqrt{c x^{2} + b x}}{1280 \, c^{4}}, \frac{15 \, b^{5} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (128 \, c^{5} x^{4} + 176 \, b c^{4} x^{3} + 8 \, b^{2} c^{3} x^{2} - 10 \, b^{3} c^{2} x + 15 \, b^{4} c\right )} \sqrt{c x^{2} + b x}}{640 \, c^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (x \left (b + c x\right )\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33394, size = 128, normalized size = 1.16 \begin{align*} \frac{3 \, b^{5} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{7}{2}}} + \frac{1}{640} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c x + 11 \, b\right )} x + \frac{b^{2}}{c}\right )} x - \frac{5 \, b^{3}}{c^{2}}\right )} x + \frac{15 \, b^{4}}{c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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